I am having the same issue, just that it didnt happen after an update, i just bought the game today. I bought it and all DLC, installed it all and when i launch origin closes, and comes back after some seconds, the game does not launch, there is no activity in task manager. I have the game only in Origin, steam is closed. I tried:
-Repairing the game
-Reinstalling the game
-Rebooting windows
-Move the license folder, a new license was created
-move the biogame folder, a new one was created
-Run the exe directly, both as admin and without
-Disable origin in game
-Disable cloud saves
I noticed that when i click on my profile in origin it says "playing: mass effect 3", so origin thinks mass effect is running.
I did install mods (just ALOT, HD texture mod), but deleted it when i reinstalled the game.
Other games in origin (ME1, ME2, the dragon age games) work, with and without mods.
Im running windows 10 Version 20H2 Build 19042,1052
EDIT: and heres my DxDiag
EDIT2: forgot to mention that its the original one, not legendary edition
EDIT3: Also tried a full clean reinstall of origin as described in post #6